Assignment_2_Solutions

Assignment_2_Solutions - Stat 371 Spring 2011 Assignment 2...

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Stat 371 Spring 2011 Assignment 2 Solutions 1. (Some fun with covariances) Appendix 2 in the course notes deals with expectation, variance and covariance for vectors of random variables. This question will give you some practice playing with the formulas. a) If U and V are two vectors of random variables of length m and n respectively, and is an [,] Cov U V mn × matrix with element , prove that th ij j i U V j Cov [ , ] [( [ ])( [ ]) ] T Cov U V E U E U V E V =− We have and the matrix is m with entry ( and so the result follows. [ , ] [( [ ])( [ ])] ij i i j j Cov U V E U E U V E V ( [ ])( [ ]) T UE UVE V −− n × th ij [ ])( [ ]) ii j V b) Show that [ , ] [ , ] CovAUV ACovUV = and [ , ] [ , ] T Cov U BV Cov U V B = Using the results in a), we have [, ][ ( [] ) ( [ ] ) [( ) ( )] [( [ ])( [ ]) ] T T T Cov AU V E AU E AU V E V EAU AEU V EV AE U E U V E V ] and [, ] [ ( ) ( [ ] [( [ ])( [ ]) ] [( [ ]){ ( [ ])} ] [( [ ])( [ ]) ] T T T TT Cov U BV E U E U BV E BV EU EU BV EBV E U EU BV EV EU EU V EV B c) For the regression model YX R β = + where 2 ~( 0 , ) R N σ I 0 , show that and ] Cov X r = ± ± [ , ] 0 Cov Y r ± 2 ] ( ) ] ( ) () 0 T Cov X r Cov HY I H Y HCov Y Y I H HI IH = ± ± since ( ) I H is symmetric. 2 ( ( ) T Cov Y r Cov Y I H Y Cov Y Y I H ± which is not a matrix of 0’s.
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2. The owners of an on-line shopping site investigated two possible changes C1 and C2 to the design of the site. In a usability trial, customers were recruited and assigned at random to one of the four possible sites (current, C1 only, C2 only, both C1 and C2). Customers in the trial were directed to order a particular item (which they received for free) and the time for the completed transaction was recorded. Participants in the trial also answered a question on a 7 point scale about how familiar they were with internet shopping. The data are in the file ass2q2.txt with variates y: time in minutes to complete the transaction x1=1 for current design, x1=0 otherwise x2=1 for C1 only, x2=0 otherwise x3=1 for C2, x3=0 otherwise x4=1 for C1 and C2, x4=0 otherwise score: answer to the question Consider the model 12 3 4 5 1234 Y xxxxs c o r e R β ββββ =+ + + + + a) Why is the intercept term excluded from the model? Note that exactly one of 1 , 2 , 3 , 4 iiii x xxx is 1 and the other three are 0 so we have . If we include an intercept term, then the columns of xx +++= 1 X will be linearly dependent. b) Give a careful interpretation of the parameter 4 We can only interpret 4 when 123 x 0 = == and 41 x = when the model becomes 45 Ys c o r e R + . Hence 4 represents the expected value of Y when the score is 0. [Not very useful since the score cannot be 0 – the parameters of interest are really the differences such as ] c) Find a 95% confidence interval for 4 .
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This note was uploaded on 09/14/2011 for the course STAT 371 taught by Professor Ahmed during the Spring '09 term at Waterloo.

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Assignment_2_Solutions - Stat 371 Spring 2011 Assignment 2...

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